TR-YÖS 2026 Derslerimiz başladı #trtös2026#yabancıögrenci

Main Ideas / Concepts Taught

• Working with radicals (square roots) and exponents

  • Rewrite the degree (index) of a root in exponential form so products and simplifications become easier.
  • Use identities such as:
    • (\sqrt{a}\cdot \sqrt{a}=a)
    • (\sqrt{m}\cdot \sqrt{n}=\sqrt{mn}) (for combining radicals)

• Rationalizing denominators with conjugates

  • If a denominator contains a radical, multiply the numerator and denominator by the conjugate to eliminate the radical.
  • Recognize that this often produces a difference of squares.

• Difference of two squares

  • For expressions of the form ((A-B)(A+B)), use:

    • [ (A-B)(A+B)=A^2-B^2 ]

  • Key emphasis: don’t distribute incorrectly—the structure determines the simplification.

• Converting and matching radical forms

  • Strategy: rewrite radicals into a common base/exponent format (sometimes using auxiliary variables like (a,b,c)) so expressions can be compared or written in a required target form.
  • Example goal: express (\sqrt{10}) “in terms of (a).”

• Ordering numbers involving exponents

  • If expressions are written with matching exponents/degrees, compare the inner quantities to determine magnitude.
  • For fractions/powers in denominators: the expression with the largest denominator (within the same structure) is typically smallest overall.
  • General rule used:
    • When denominators share the same “degree structure,” larger denominator → smaller fraction.

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Detailed Methodologies / Step-by-Step Instruction Segments (as implied)

1) Subtracting and simplifying radical expressions (example logic shown)

1. Rewrite radical indices using exponents so terms like (\sqrt{x}) are handled consistently.
2. Convert terms so products of like radicals become integers, e.g.:
   • (\sqrt{2}\cdot\sqrt{2}=2)
   • (\sqrt{5}\cdot\sqrt{5}=5)
   • (\sqrt{3}\cdot\sqrt{3}=3)
3. When subtracting or canceling radicals:
   • convert to like terms first, then combine coefficients.

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2) Rationalizing a denominator using conjugates + difference of squares

1. Identify conjugate forms:
   • If you see ((\sqrt{u}\pm \sqrt{v})), pair it with ((\sqrt{u}\mp \sqrt{v})).
2. Multiply the expression by the conjugate.

3. Apply the difference-of-two-squares pattern:

   • [ (\sqrt{u}-\sqrt{v})(\sqrt{u}+\sqrt{v})=u-v ]

4. Expand carefully:

   • outer signs produce (u-v)
   • cross terms cancel due to conjugation.

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3) Converting radicals into a target parameter form (e.g., “in terms of (a)”)

1. Decompose each radical into products of simpler radicals that match the target structure.
   • Example patterns used:
     • (\sqrt{6}=\sqrt{3}\sqrt{2})
     • (\sqrt{15}=\sqrt{3}\sqrt{5})
2. Use given relationships to rewrite pieces (example stated):
   • (\sqrt{2}=\dfrac{b}{\sqrt{3}})
3. Re-express everything so the final result matches the required format (as suggested by the subtitles), such as:
   • a form like (\dfrac{B\cdot C}{A^2}) (i.e., “in terms of (a)” style formatting)
4. Avoid confusing or disordered distribution when manipulating exponents.

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4) Ordering numbers from smallest to largest (power/denominator logic)

1. If exponent structure matches across expressions:
   • compare inner values to determine magnitude (larger inner base with matched exponents ⇒ larger overall value).
2. If expressions are fractions with a common pattern:
   • larger denominator ⇒ smaller fraction.
3. Example logic described:
   • denominators like (3^k) were compared by rewriting/recognizing shared exponent bases,
   • then smallest/largest was determined by which denominator exponent is comparatively larger/smaller.

Core takeaway: consistency of exponent/degree structure is what makes ordering possible.

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Speakers / Sources Mentioned (from subtitles)

• Teacher (spoken to students; no specific name given)
• Professor (referred to as a speaker/authority; name not given)
• Student(s) — e.g., Aykut (explicitly named)